Quantum Mechanics — Lecture notes for PHYS223 XX Dirac notation for composite systems

XXI Many particles

Quantum mechanics takes one more twist when we consider a system containing many indistinguishable particles. By this we many particles that share the same physical characteristics like mass, charge, overall spin, etc; examples are collections of many electrons, or even collections of many identical atoms (which themselves are composite systems).

Assume, therefore, that there are several quantum particles, labelled $1,2,3,\ldots$ , which can be found at positions ${\bf r_{1}}$ , ${\bf r_{2}}$ , ${\bf r_{3}}$ , $\ldots$ . Some of the particles may also have spin $s_{1}$ , $s_{2}$ , $s_{3}$ , $\ldots$ . The spin can take values $s_{i}=m\hbar$ where $m=-S_{i},-S_{i}+1,\ldots,S_{i}$ . Particles are called bosons when $S_{i}$ is an integer (an example is the photon, with $S=1$ ). Particles are called fermions when $S_{i}$ is a halfinteger (i.e., an integer plus 1/2). Examples are electrons and protons, which have $S=1/2$ .

XXI.1 Distinguishable particles

For a collection of distinguishable particles, we can follow the description of systems with many degrees of freedom. The total wavefunction is given by $\psi({\bf r_{1}},s_{1},{\bf r_{2}},s_{2},{\bf r_{3}},s_{3},\ldots)$ . An example is the wave function $\psi({\bf r_{p}},s_{p},{\bf r_{e}},s_{e})$ describing the proton and the electron in a hydrogen atom.

XXI.2 Indistinguishable particles

For indistinguishable particles, the total wavefunction is still a function $\psi({\bf r_{1}},s_{1},{\bf r_{2}},s_{2},{\bf r_{3}},s_{3},\ldots)$ , but probabilities cannot depend on the order of the labels:

|\psi({\bf r_{1}},s_{1},{\bf r_{2}},s_{2},{\bf r_{3}},s_{3}\ldots)|^{2}=|\psi(% {\bf r_{2}},s_{2},{\bf r_{1}},s_{1},{\bf r_{3}},s_{3},\ldots)|^{2},

(475)

and similarly for the interchange of the other labels. This leaves two options. The first option is

\psi({\bf r_{1}},s_{1},{\bf r_{2}},s_{2},{\bf r_{3}},s_{3}\ldots)=\psi({\bf r_% {2}},s_{2},{\bf r_{1}},s_{1},{\bf r_{3}},s_{3},\ldots),

(476)

etc: The wavefunction is symmetric. This case is found when the particles are bosons. The second option is

\psi({\bf r_{1}},s_{1},{\bf r_{2}},s_{2},{\bf r_{3}},s_{3}\ldots)=-\psi({\bf r% _{2}},s_{2},{\bf r_{1}},s_{1},{\bf r_{3}},s_{3},\ldots),

(477)

etc: The wavefunction is antisymmetric. This case is found when the particles are fermions.

XXI.3 Pauli exclusion principle

In the case of fermions, Eq. (477) implies that the probability to find two particles at the same point in space is zero if their spin state is the same:

\psi({\bf r_{1}},s_{1},{\bf r_{1}},s_{1},{\bf r_{3}},s_{3}\ldots)=-\psi({\bf r% _{1}},s_{1},{\bf r_{1}},s_{1},{\bf r_{3}},s_{3},\ldots),

(478)

hence

\psi({\bf r_{1}},s_{1},{\bf r_{1}},s_{1},{\bf r_{3}},s_{3}\ldots)=0.

(479)

This is called the Pauli exclusion principle, which has numerous important consequence ranging from the allowed configurations of occupied orbitals by electrons in an atom and the periodic table of elements (both discussed below) over the general structure of matter to the properties of neutron stars.

XXI.4 Two-electron orbitals

The wavefunction of two electrons can be grouped into a vector

\psi=\left(\begin{array}[]{c}\psi({\bf r_{1}},\uparrow,{\bf r_{2}},\uparrow)\\ \psi({\bf r_{1}},\downarrow,{\bf r_{2}},\uparrow)\\ \psi({\bf r_{1}},\uparrow,{\bf r_{2}},\downarrow)\\ \psi({\bf r_{1}},\downarrow,{\bf r_{2}},\downarrow)\\ \end{array}\right).

(480)

Typical two-electron wavefunctions which satisfy Eq. (477) are of the form

[\psi_{1}({\bf r_{1}})\psi_{2}({\bf r_{2}})-\psi_{1}({\bf r_{2}})\psi_{2}({\bf r% _{1}})]\left(\begin{array}[]{c}1\\ 0\\ 0\\ 0\\ \end{array}\right),

(481)

[\psi_{1}({\bf r_{1}})\psi_{2}({\bf r_{2}})-\psi_{1}({\bf r_{2}})\psi_{2}({\bf r% _{1}})]\left(\begin{array}[]{c}0\\ 1/\sqrt{2}\\ 1/\sqrt{2}\\ 0\\ \end{array}\right),

(482)

and

[\psi_{1}({\bf r_{1}})\psi_{2}({\bf r_{2}})-\psi_{1}({\bf r_{2}})\psi_{2}({\bf r% _{1}})]\left(\begin{array}[]{c}0\\ 0\\ 0\\ 1\\ \end{array}\right).

(483)

In all these cases the orbital part is antisymmetric and the spin part is symmetric. These states are called triplet states and correspond to a total spin of $\hbar$ (i.e. the spin of the two electrons is parallel and adds up). There is only a single state where the spin is anti-parallel:

[\psi_{1}({\bf r_{1}})\psi_{2}({\bf r_{2}})+\psi_{1}({\bf r_{2}})\psi_{2}({\bf r% _{1}})]\left(\begin{array}[]{c}0\\ 1/\sqrt{2}\\ -1/\sqrt{2}\\ 0\\ \end{array}\right).

(484)

This state corresponds to a total spin of 0 and has a symmetric orbital part.

When the orbital wavefunctions are identical, $\psi_{1}=\psi_{2}$ , then the triplet states vanish. This is a consequence of the Pauli exclusion principle: Two electrons cannot have the same orbital wavefunction if their spin is parallel.

XXI.5 Two electrons in a harmonic-oscillator potential

Recall that the single-particle states in a harmonic-oscillator potential are enumerated by a quantum number $|n\rangle$ , $n=0,1,\ldots,$ , and associated with an energy $E_{n}=\hbar\omega(n+1/2)$ . If we load electrons into such a harmonic trap we also have account for the spin, degree of freedom, thus resulting in states $|n,\uparrow\rangle$ and $|n,\downarrow\rangle$ whose energies are degenerate, $E_{n,\uparrow}=E_{n,\downarrow}=E_{n}$ .

For two electrons, we now have to distinguish several situations:

If the spins are identical, e.g., pointing up, we then have to work in the basis of states $|n\uparrow;m\uparrow\rangle$ . The spin part of this is symmetric, and thus the antisymmetry required by the Pauli principle must arise from the orbital part, giving rise to a structure

|n\uparrow;m\uparrow\rangle-|m\uparrow;n\uparrow\rangle=(|nm\rangle-|mn\rangle% )\mid\uparrow\uparrow\rangle,

(485)

where we formally separated the orbital and spin parts of the state. In terms of wave functions, the orbital part is of the structure $\psi_{n}(x_{1})\psi_{m}(x_{2})-\psi_{m}(x_{1})\psi_{n}(x_{2})$ , where $x_{i}$ are the coordinates of the two electrons. This state vanishes if $n=m$ . In particular, the lowest-energy state of the system is $(|01\rangle-|10\rangle)\mid\uparrow\uparrow\rangle$ ; this has energy $E_{01}=E_{0}+E_{1}=2\hbar\omega$ .

If the spins are non-identical, we have to work in the basis of states $|n\uparrow;m\downarrow\rangle$ and $|n\downarrow;m\uparrow\rangle$ . The spin part may still be symmetric, which is realised by states of the form

	$\displaystyle\|n\uparrow;m\downarrow\rangle+\|n\downarrow;m\uparrow\rangle-\|m% \uparrow;n\downarrow\rangle-\|m\downarrow;n\uparrow\rangle$
	$\displaystyle=(\|nm\rangle-\|mn\rangle)(\mid\uparrow\downarrow\rangle+\mid% \downarrow\uparrow\rangle).$		(486)

Again, these states vanish for $n=m$ . This exhausts the triplet states, which are all constrained by the Pauli principle. However, the spin part may now also be antisymmetric, which is realised by states of the form

	$\displaystyle\|n\uparrow;m\downarrow\rangle-\|n\downarrow;m\uparrow\rangle+\|m% \uparrow;n\downarrow\rangle-\|m\downarrow;n\uparrow\rangle$
	$\displaystyle=(\|nm\rangle+\|mn\rangle)(\mid\uparrow\downarrow\rangle-\mid% \downarrow\uparrow\rangle).$		(487)

These are singlet states, for which $n$ and $m$ may be equal. This includes the lowest-energy state $|00\rangle(\mid\uparrow\downarrow\rangle-\mid\downarrow\uparrow\rangle)$ , which possesses energy $E_{00}=2E_{0}=\hbar\omega$ .

XXI.6 The periodic table of chemical elements

The periodic table of elements is composed of atoms with a nucleus of charge $Z e$ , which attracts $Z$ electrons. A qualitative understanding can be obtained by ignoring the interaction between the electrons. Then the electronic states are given by the states $\psi_{nml}=R_{n}Y_{lm}$ of the hydrogen atom, but with the charge $e$ of the nucleus replaced by $Z e$ . The ground state of an atom is obtained by occupying these states starting with the lowest energy state $1s$ , then $2s$ and $2p$ , then $3s$ , $3p$ . (After this point interactions start to change the systematics slightly: The 4s orbital is filled before 3d.) Because of the Pauli principle, each state can accommodate only two electrons, which have to have opposite spin.

Examples: Helium (He) has $Z=2$ (the nucleus consists of two protons and two neutrons). Hence, it has two electrons, which occupy the 1s orbital with opposite spin. The electronic charge density is spherical symmetric, and exciting an electron into the 2s or 2p orbital would cost a relatively large amount of energy. This explains why He is chemically inert.

Neon (Ne) has $Z=10$ , hence all states in 1s, 2s and 2p are occupied. Again, the charge distribution is spherical symmetric and the excitation energy to the 3s, 3p and 3d orbitals prevents this atom from being chemically active. Atoms of similar properties as He or Ne (noble gases) are all found in the last column of the periodic table.

Sodium (Na) has $Z=11$ . Ten electrons occupy the 1s, 2s, and 2p orbitals, like in Ne, and the extra eleventh electron occupies the 3s orbital. Its chemical properties are typical for the alkali metals, which are all found in the first column of the periodic table, and all have the electronic structure of a noble gas plus one extra electron in an s orbital. Alkali atoms like to bind to halogens, the atoms in the last-but-one column, which lack one electron for a spherically symmetric electronic configuration. When an alkali atom binds to a halogen atom, the alkali atom donates and electron to the halogen atom, which is energetically favorable; both atoms then are spherically symmetric and oppositely charged, and attract each other by the Coulomb force.

XXI.7 Quantum statistical mechanics ${}^{*}$

We finally turn to a brief discussion of the consequences of quantum mechanics for statistical mechanics, which concerns the thermodynamical description of large composite systems in (or close) to equilibrium. This topic brings us right back to the historical origin of quantum mechanics: Max Planck’s hypothesis that the energy of systems can be quantised. That we can discuss these consequences solidly only at the end of this course is testament to the depth of this original concept (and well justifies the appellation quantum mechanics for the theory that emerged from it).

Thermodynamics enjoys a firm theoretical foundation relying on statistical considerations of the occupation probability of all possible microstates, as pioneered by Ludwig Boltzmann, who introduced statistical weights which relate the occupation probability to energy and temperature. Quantum mechanics fundamentally affects and shapes all aspects of this description via the concepts of energy quantisation, rules to calculate occupation probabilities, and the Pauli principle which implies restrictions for the states of indistinguishable particles, of which there are two variants, bosons and fermions.

XXI.7.1 Boltzmann distribution

The Boltzmann distribution assigns the value

P_{n}=\frac{1}{Z(T)}g_{n}\exp(-E_{n}/kT)

(488)

to the probability that a thermodynamical system occupies a microstate of energy $E_{n}$ and degeneracy $g_{n}$ , where $k$ is Boltzmann’s constant, $T$ is temperature, and

Z(T)=\sum_{n}g_{n}\exp(-E_{n}/kT)

(489)

is the partition function which ensures normalisation of the occupation probabilities.

Quantum mechanically, this corresponds to a density matrix

\hat{\rho}=\frac{1}{Z(T)}\exp(-\hat{H}/kT),

(490)

where now normalisation enforces $Z(T)={\rm tr}\,\hat{\rho}$ .

XXI.7.2 Fermi-Dirac statistics

Consider a thermodynamical systems which is made out of identical components. Starting from the Boltzmann distribution of the combined system we can derive probability distributions for states individual components.

The simplest case concerns fermions, since here each single-particle state $|i\rangle$ can be occupied at most with probability one. In a measurement, this corresponds to only two choices: The state is occupied or empty. Based on the Boltzmann distribution for the microstates of the total system, the average number of fermions occupying this then follows the Fermi-Dirac distribution

\bar{n}_{i}=\frac{1}{\exp[(\varepsilon_{i}-\mu)/kT]+1},

(491)

where $\mu$ is the chemical potential, which is independent of the index $i$ and can be determined by knowledge of the total number $N=\sum_{i}\bar{n}_{i}$ of particles in the system.

At low temperatures, the Fermi-Dirac distribution behaves like a unit step function, $\bar{n}_{i}\approx\Theta(E_{F}-\varepsilon_{i})$ , where $E_{F}$ is the Fermi energy. In many applications, the Fermi energy is very large. An important example are electrons in metals, which typically have a Fermi energy $E_{F}$ is many thousands of kelvins). Electrons in a metal can be treated as weakly interacting, and therefore are a good approximation of a (Fermi gas). For $E_{F}\ll T$ (i.e., at room temperature and below), the gas is called degenerate.

Further examples of degenerate Fermi gases are neutron stars (a degenerate gas of neutrons) and white dwarf stars (a degenerate gas of electrons). A degenerate gas is difficult to compress since the density of states decreases with volume, upon which the Pauli principle enforces occupation of highly energetic levels. This corresponds to a large degeneracy pressure which stabilizes white dwarfs and neutron stars against gravitational collapse as long as they are not too heavy. (White dwarfs are stable below the Chandrasekhar limit of about 1.4 solar masses; exceeding this limit they may explode in a Type Ia supernova. Neutron stars are formed as remnants of Type II, Type Ib or Type Ic supernovae, involving more massive stars. Above 3-4 solar masses, the degeneracy pressure of the neutrons is overcome by gravity, which may lead to the formation of a black hole.)

XXI.7.3 Bose-Einstein statistics

Bosons are not subject to the Pauli exclusion principle, so that each single-particle state can be occupied an arbitrary number of times. The Boltzmann distribution for the microstates of the total system then yields the Bose-Einstein distribution

\bar{n}_{i}=\frac{1}{\exp[(\varepsilon_{i}-\mu)/kT]-1},

(492)

which differs from the Fermi-Dirac distribution only by a $-$ sign in the denominator.

At low temperatures, the chemical potential of a weakly interacting Bose gas tends to zero. Below a critical temperature, the ground state becomes populated by finite fraction of the total number of boson, and the gas forms a Bose-Einstein condensate, as first realised in 1995 by Eric Cornell, Carl Wieman, and co-workers with atomic gases at sub-micro Kelvin temperatures. Analogously, more strongly interacting bosons may form a superfluid with zero viscosity, as first observed for Helium-4 (by Pyotr Kapitsa, John Allen and Don Misener, in 1938). Under some circumstances, strongly bound pairs of fermions can effectively behave like bosons. These can condense to form fermionic superfluids, like a superfluid of electronic Cooper pairs in a superconductor (as proposed by John Bardeen, Leon Cooper and Robert Schrieffer in 1957; superconductivity was discovered by Heike Kamerlingh Onnes in 1911), or a superfluid of Helium 3 (as confirmed by Douglas D. Osheroff, in 1971).

XXI.7.4 Maxwell-Boltzmann statistics

At large temperatures and low densities, the Fermi-Dirac distribution and the Bose-Einstein distribution tend to the same limit, the Maxwell-Boltzmann distribution

\bar{n}_{i}=\exp[-(\varepsilon_{i}-\mu)/kT].

(493)

For a gas, this approximation is valid if its particle concentration $c$ is much smaller than the quantum concentration $c_{Q}=(mkT/2\pi\hbar^{2})^{3/2}$ . This defines the limit of a non-degenerate gas.

XXI.7.5 Planck’s law of black body radiation

As an application we consider the problem of electromagnetic radiation intensity emitted by a black body, an object that perfectly absorbs all incident electromagnetic radiation. Classical treatment of this radiation leads to two contradictory predictions, Wien’s law (which is only accurate at high frequencies), and the Rayleigh-Jeans law (which is only accurate at low frequencies). This impasse led Planck to formulate his quantisation hypothesis. The problem is then solved by combining the density of electromagnetic field states

\rho(\omega)=L^{3}\frac{\omega^{2}}{\pi^{2}c^{3}}

(494)

per unit angular frequency interval with the occupation probability

\bar{n}(\omega)=\frac{1}{\exp[\hbar\omega/kT]-1}

(495)

of photons with energy $\hbar\omega$ . This delivers an energy density

u(\omega,T)=\frac{1}{L^{3}}\hbar\omega\rho(\omega)\bar{n}(\omega)=\frac{\hbar% \omega^{3}}{\pi^{2}c^{3}}\frac{1}{\exp[\hbar\omega/kT]-1}.

(496)

The direction-resolved spectral intensity per unit solid angle is given by

I(\omega,T)=\frac{cu(\omega)}{4\pi}=\frac{\hbar\omega^{3}}{4\pi^{3}c^{2}}\frac% {1}{\exp[\hbar\omega/kT]-1}.

(497)

In terms of the ordinary frequency $\nu=\omega/2\pi$ , this delivers Planck’s law

I(\nu,T)=2\pi I(\omega=2\pi\nu,T)=\frac{2h\nu^{3}}{c^{2}}\frac{1}{\exp[h\nu/kT% ]-1},

(498)

where we introduced (for good measure) Planck’s original constant $h=2\pi\hbar=6.62606896(33)\times 10^{-34}{\rm J\,s}$ .